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Theorem grpss 14503
Description: Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring  R is a group before we know that it is also a ring. (Theorem rnggrp 15346, on the other hand, requires that we know in advance that  R is a ring.) (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
grpss.g  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
grpss.r  |-  R  e. 
_V
grpss.s  |-  G  C_  R
grpss.f  |-  Fun  R
Assertion
Ref Expression
grpss  |-  ( G  e.  Grp  <->  R  e.  Grp )

Proof of Theorem grpss
StepHypRef Expression
1 grpss.r . . . 4  |-  R  e. 
_V
2 grpss.f . . . 4  |-  Fun  R
3 grpss.s . . . 4  |-  G  C_  R
4 baseid 13190 . . . 4  |-  Base  = Slot  ( Base `  ndx )
5 opex 4237 . . . . . 6  |-  <. ( Base `  ndx ) ,  B >.  e.  _V
65prid1 3734 . . . . 5  |-  <. ( Base `  ndx ) ,  B >.  e.  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. }
7 grpss.g . . . . 5  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. }
86, 7eleqtrri 2356 . . . 4  |-  <. ( Base `  ndx ) ,  B >.  e.  G
91, 2, 3, 4, 8strss 13183 . . 3  |-  ( Base `  R )  =  (
Base `  G )
10 plusgid 13243 . . . 4  |-  +g  = Slot  ( +g  `  ndx )
11 opex 4237 . . . . . 6  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  _V
1211prid2 3735 . . . . 5  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. }
1312, 7eleqtrri 2356 . . . 4  |-  <. ( +g  `  ndx ) , 
.+  >.  e.  G
141, 2, 3, 10, 13strss 13183 . . 3  |-  ( +g  `  R )  =  ( +g  `  G )
159, 14grpprop 14501 . 2  |-  ( R  e.  Grp  <->  G  e.  Grp )
1615bicomi 193 1  |-  ( G  e.  Grp  <->  R  e.  Grp )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {cpr 3641   <.cop 3643   Fun wfun 5249   ` cfv 5255   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489
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